As mentioned in the comment by Jeff Zeitlin above, this is a good match to Michael Kube-McDowell's Alternities, published 1988.
As taken from the outline posted by user Scott Halstad at goodreads.com, some main characters are:
Rayne Wallace, a "Runner" who goes between alternate versions of America...
President Robinson, who's a psychotic intent upon starting a nuclear war with Russia, which in this world (the "Home" alternity) is
a big bully to pussycat America... he intends to use these alternate
Earths as escape vehicles for he and his government cronies so that
they can continue to dominate worlds while their America is
obliterated by Russian nukes...
Senator Endicott, who discovered the "gates" to these alternative
Americas, although we're never told how. He has women from these
alternities brought over for him to serve as sex slaves whom he
ultimately murders...
Halstad's review also mentions a significant plot point that may ring a bell:
Then there's the mysterious maze that lies between the alternate gates with its own demon that destroys people it encounters.
I've also read the book, and I can confirm that it ends on the note that a new reality has been identified, which is implied to be our own. I found this review, which contains an aside that confirms Jimmy Carter is mentioned:
(our own timeline is discovered late in the novel, and merits only a
brief line about President Carter)
As I recall, it is also discovered there are a limited number of alternities, and that the shape of the maze is a regular polyhedron (dodecahedron? icosahedron?). Also, within the main character's organization, each alternity is designated by a color, and most of the action centers around Alternity Red and Alternity Blue.
